You just wrote the recipe that will fix your team. Not every talented player wants to play like Wendell Clark or Doug Gilmour so you need bottom six guys that can counter opposing teams when the going gets tough. A guy like Jeannot could easily be moved up as protection for one of your top 4 guys if need be. In the playoffs a team needs to be able to win games in different ways but allowing your talented guys to be intimidated without any answer is a recipe for failure.
When we (habs) eliminated the Leafs in 21, it certainly wasn't because we could match you guys with talent or firepower. Carey Price was good yes but it was Weber, Chiarot and Perry that got in the heads of your best players and doubt set in and the series was lost.
Shanahan was a tough, gritty, talented player that won cups so it's confusing to many as to why the Leafs don't seem to value guys that contribute more than just points as you said.